A 61.0-kg runner has a speed of 3.20 m/s at one instant during a long-distance event.

(a) What is the runner's kinetic energy at this instant?
KEi = J

(b) If he doubles his speed to reach the finish line, by what factor does his kinetic energy change?
KEf
KEi
=

Think I answered this once

KE = 1/2 mv^2, so just plug in your numbers.

Now, if we have 2v instead of v, that gives us

1/2 m(2v)^2 = 1/2 m*4v^2 = 4(1/2 mv^2)

since KE varies as v^2, 2v gives 2^2 times the energy.

To calculate the kinetic energy of a runner, we can use the equation:

Kinetic Energy (KE) = 1/2 * mass * velocity^2

(a) To find the runner's kinetic energy at this instant, we plug in the given values into the equation:

Mass (m) = 61.0 kg
Velocity (v) = 3.20 m/s

KE = 1/2 * 61.0 kg * (3.20 m/s)^2

Simplifying the equation:

KE = 1/2 * 61.0 kg * 10.24 m^2/s^2
KE = 312.32 J

Hence, the runner's kinetic energy at this instant is 312.32 Joules.

(b) If the runner doubles his speed, we need to calculate the new kinetic energy (KEf) and compare it to the initial kinetic energy (KEi).

Doubling the speed means the new velocity (vf) becomes:

vf = 2 * vi
vf = 2 * 3.20 m/s
vf = 6.40 m/s

Now, substitute the new mass and velocity into the kinetic energy equation:

Mass (m) = 61.0 kg
Velocity (v) = 6.40 m/s

KEf = 1/2 * 61.0 kg * (6.40 m/s)^2

Simplifying the equation:

KEf = 1/2 * 61.0 kg * 40.96 m^2/s^2
KEf = 1250.56 J

The kinetic energy (KEf) when the runner doubles his speed is 1250.56 Joules.

To find the factor by which the kinetic energy changes, we can divide the final kinetic energy (KEf) by the initial kinetic energy (KEi):

KEf / KEi = 1250.56 J / 312.32 J
KEf / KEi ≈ 4

Therefore, when the runner doubles his speed, his kinetic energy increases by a factor of approximately 4.